22 We follow the tie-breaking assumption of Sec. Hb that when indifferent the indi- vidual adopts.
Proposition 3. If there is a probability, bounded from zero, of further public information release before each individual chooses (and the public information is conditionally independent and identically distributed and assumptions 1 and 2 are satisfied), then individuals eventually settle into the correct cascade.
The proof is a standard application of the law of large numbers and is omitted. It relies on the fact that as the number of public information releases increases, the correct choice becomes clearer. The strong law of large numbers ensures that as long as public information is conditionally independent and identically distributed, the posterior concentrates on the true value and each individual almost surely decides correctly. Thus each individual acts like the previous individual, and the correct cascade results.
Since proposition 3 relies on asymptotic arguments, it provides only moderate grounds for optimism. Further intuition can be gained from a numerical example. Consider again the binary-signal/value case discussed in Section 11 A. However, we now introduce a small probability that an information signal, drawn independently from the same distribution as each individual's signal, is publicly released. Columns 1—2, 4-5, and 7—8 in table 2 list the probabilities that an up cascade and a down cascade will be in process when the 1,000th individual is reached as a function of p, the probability that the signal is H given that the actual value of V is one. The probability of settling into the correct up cascade increases dramatically even when only very few public releases of information occur on average. For example, if p = .75, the probability of ending up in the correct cascade increases from .81 when there is no public information release to .86 (.98) when on average one (10) release(s) of public information occurs per 1,000 individuals.
As a possible case in which an incorrect cascade started and then reverted to the correct cascade, Apodaca (1952) documented the introduction of one variety of hybrid seed corn for 84 growers in a New Mexico village from 1945 to 1949 in which a trend reversed before settling on an outcome. Since the hybrid seed yielded three times as much as the old seed, the percentage of adopters rose from 0 percent in 1945 to 60 percent in 1947. However, 2 years later it fell back to 3 percent when the villagers decided that the hybrid corn tasted worse.
Columns 3,6, and 9 record the expectation of the difference between the number of inferred H and L signals after 1,000 individuals. With public information, the average cascade is quite deep and correctly positive. If p = .65 (a rather noisy signal), with 10 per 1,000 public information releases, the expected difference is 4.6.
Table 2
Effect
of Multiple Information Releases
B. Discussion
The social cost of cascades is that the benefit of diverse information sources is lost. Thus a cascade regime may be inferior to a regime in which the actions of the first n individuals are observed only after stage n + 1. However, the fragility of cascades allows some of the benefit of information diversity to be recaptured. Incorrect decisions, once taken, can be rapidly reversed. For instance, a high-precision individual late in the sequence can break a cascade, which leads to better decisions. Public information disclosures can break incorrect cascades and eventually bring about the correct decision.